Devices, Systems, And Methods Employing Polynomial Symbol Waveforms

ABSTRACT

Systems, devices, and methods of the present invention enhance data transmission through the use of polynomial symbol waveforms (PSW) and sets of PSWs corresponding to a symbol alphabet is here termed a PSW alphabet. Methods introduced here are based on modifying polynomial alphabet by changing the polynomial coefficients or roots of PSWs and/or shaping of the polynomial alphabet, such as by polynomial convolution, to produce a designed PSW alphabet including waveforms with improved characteristics for data transmission.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.17/577,082 filed 2022 Jan. 17, now U.S. patent Ser. No. 11/729,041issued 2023 Aug. 15, which is a continuation of U.S. patent applicationSer. No. 17/060,181 filed 2020 Oct. 1, now U.S. Pat. No. 11,228,477issued 2022 Jan. 18, which is a continuation of U.S. patent applicationSer. No. 16/735,655 filed 2020 Jan. 6, now U.S. Pat. No. 10,848,364issued 2020 Nov. 24, which claims the benefit of and priority from U.S.Provisional Patent Application No. 62/814,404, filed 2019 Mar. 6,entitled “Devices, Systems, And Methods Employing Polynomial SymbolWaveforms”, each of which is incorporated herein by reference in itsentirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under Award #1738453awarded by the National Science Foundation. The government has certainrights in the invention.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates in general to transmitting data, and, morespecifically to systems, devices, and methods employing polynomialsymbol waveforms to enable higher capacity data transmission.

Background Art

Applicant's prior U.S. Pat. No. 8,472,534 entitled “TelecommunicationSignaling Using Non-Linear Functions” and U.S. Pat. No. 8,861,327entitled “Methods and Systems for Communicating”, the contents of whichare herein incorporated by reference in their entirety, introducedspiral-based signal modulation. Spiral-based signal modulation may basesignal modulation on complex spirals, rather than the traditionalcomplex circles used by standard signal modulation techniques such asQuadrature Amplitude Modulation (QAM) and Phase-Shift Keying (PSK). Theuse of such advanced modulation formats enables higher capacity datatransmission systems, devices, and methods.

Applicant's U.S. Pat. No. 10,069,664 entitled “Spiral PolynomialDivision Multiplexing” (SPDM), the contents of which are hereinincorporated by reference in their entirety, discloses the use ofpolynomials to describe signals and “Instantaneous Spectral Analysis”(ISA) to decompose arbitrary polynomials, which may represent a signal,into a sum of complex spirals, which may have a real-valuedrepresentation as a sum of sinusoids, each of which may have acontinuously-varying amplitude. ISA introduced a method for identifyingand grouping sinusoids with the same frequency together fortransmission, such that the amplitude and power associated with eachsinusoidal frequency at each instant in time in the spiralrepresentation of a particular polynomial may be readily apparent.

With the continuing demand for high capacity data transmission systems,devices, and methods, there is a corresponding need for systems,devices, and methods that enable data to be more efficientlycharacterized for transmission.

BRIEF SUMMARY OF THE INVENTION

Systems, devices, and methods of the present invention enable more datatransmission through the use of polynomial symbol waveform (PSW) designby modifying a polynomial alphabet by changing the polynomialcoefficients or roots of the PSWs and/or shaping of the polynomialalphabet, such as by polynomial convolution, to produce a designed PSWalphabet including waveforms with improved characteristics for datatransmission. The present invention may be employed with SPDM and ISA,as well as other polynomial design techniques.

A set of PSWs corresponding to a symbol alphabet is here termed a PSWalphabet. In various embodiments, transmitter and receivers mayimplement PSW alphabets produced via various PSW design methods that mayinclude specifying the location of particular polynomial roots, such as:placing roots at the symbol time boundaries with amplitude zero tominimize symbol boundary discontinuities; translating the nearestpolynomial root to a particular point in the complex plane; directlyediting the location of one or more polynomial roots; adjusting thecomplex conjugate of an edited PSW root in order to keep the PSWreal-valued; and shaping one of more PSWs using polynomial convolutionwith polynomial versions of the raised cosine, root raised cosine,Gaussian, or other pulse-shaping or band-limiting filters.

PSWs may provide communication benefits which may include greater noiseresistance and greater data throughput, particularly when coupled withISA-based transmission techniques to limit occupied bandwidth. Thisapplication introduces a coordinated set of methods to design PSWs toexploit these benefits within communication systems and devicesincluding satellite and terrestrial wireless and wired systems.

In various embodiments, systems, transmitters, receivers, and methods ofthe present invention may employ manipulated polynomial symbol waveformalphabets designed using various stochastic methods, such as, but notlimited to, Monte Carlo (MC) techniques, for optimization. The MCoptimization may be started with an initial PSW alphabet, from which itgenerates random variations (“offsets”), referred to as “Offset MC”, ora completely random PSW alphabet (“Fully random MC”). The MC routinesmay be based on polynomial roots or polynomial coefficients and may usepolynomial convolution or specify fixed polynomial roots not altered bythe stochastic method.

In various embodiments, Fractional Cycle Modulation (FCM), disclosed inthe present invention, may be performed by constructing symbol waveformsfrom sinusoids some or all of which complete a fractional cycle (lessthan a full cycle) during each symbol time. FCM-based PSWs may beemployed as the input, or initial, PSW alphabet for offset MCoptimization. Convolution or other polynomial manipulations may beapplied to an FCM-based PSW alphabet as well.

Various methods, devices including transmitters and receivers,non-transitory computer readable medium carrying instructions to beexecuted by one or more processors, and/or systems of the presentinvention may include providing, to a transmitter, an input bitsequence, converting, by the transmitter, the input bit sequence into asequence of polynomial symbol waveforms (PSWs) selected from a designedPSW alphabet, and transmitting, by the transmitter, the sequence ofpolynomial symbol waveforms. The designed PSW alphabet may be formed byproviding an initial PSW alphabet having polynomial coefficients andpolynomial roots, modifying the initial PSW alphabet to produce thedesigned PSW alphabet. In various embodiments, Instantaneous SpectralAnalysis (ISA) is performed on the sequence of polynomial symbolwaveforms prior to transmission.

Various methods, devices, non-transitory computer readable mediumcarrying instructions, and/or systems may include modifying the PSWalphabet may include at least one of editing at least one of thepolynomial coefficients or polynomial roots of the initial PSW alphabetto produce one of an edited PSW alphabet and the designed PSW alphabet,and shaping one of the edited PSW alphabet and initial PSW alphabet toproduce the designed PSW alphabet. The initial PSW alphabet may be basedon Fractional Cycle Modulation and/or other methods as described herein.Editing the PSW alphabet may include, translating at least onepolynomial root from one or more PSWs of the PSW alphabet from astarting location to an end location in the complex plane, adjustingcomplex conjugates of translated roots that are complex to keep thepolynomial real-valued, and shaping at least one of the translated andadjusted PSWs using polynomial convolution to provide the designed PSWalphabet. The end location for the translated polynomial roots are atthe symbol time boundaries with amplitude zero. Shaping may be performedby convoluting the edited PSW with a polynomial representation of afilter. The polynomial convolution is performed using one of raisedcosine, root raised cosine, Gaussian, or other band-limiting orpulse-shaping filters.

Various methods, devices, non-transitory computer readable mediumcarrying instructions, and/or systems may include modifying by applyingrandom variation to either the polynomial coefficients or polynomialroots of an initial PSW alphabet to produce an edited PSW alphabet, andmay include calculating a goodness measure for the initial and editedPSW alphabets, comparing the goodness measure of the initial PSWalphabet to the goodness measure of the edited PSW alphabet, and settingthe initial PSW alphabet equal to the edited PSW alphabet when theedited PSW alphabet has a higher goodness measure. The random variationmay be determined by a Monte Carlo optimization, machine learning, etc.A power normalization of the edited PSW alphabet may be performed afterapplying the random variation. The initial PSW alphabet may be set tothe edited PSW alphabet when the goodness measure of the edited PSWalphabet is better than the goodness measure of the initial PSWalphabet.

The steps of editing, calculating, performing, comparing, and settingmay be repeated for a user determined number of iterations and/or untilthe goodness measure of the edited PSW alphabet is not better than theinitial PSW alphabet goodness measure for one or more iterations. Thegoodness measure for a PSW alphabet may be based on calculating theminimum Root Mean Square (RMS) separation between all pairs of PSWs witha higher minimum RMS separation being interpreted as a higher goodnessmeasure. Converting the PSW alphabet may include performing a tablelookup of the input bit sequence to determine a corresponding PSW.

The sequence of PSWs transmitted by the transmitter via a wirelessand/or wired medium may be received by a receiver and the receivedsequence of PSWs may be converted to an output bit sequence, which mayexit the receiver and system and/or be processed and transmittedfurther. Converting the received sequence of PSWs may include performinga table lookup of the received sequence of PSWs to determine acorresponding output bit sequence and may include selecting the outputbit sequence that corresponds to PSW that corresponds most closely tothe received sequence of PSWs. Receiving the signal may includerecognizing the received sequence of PSWs using minimum distance signaldetection.

As may be disclosed, taught, and/or suggested herein to the skilledartisan, the present invention addresses the continuing need forhardware and/or software systems, devices, and methods that enableincreased data transmission capacity.

BRIEF DESCRIPTION OF THE DRAWINGS

Advantages of embodiments of the present invention will be apparent fromthe following detailed description of the exemplary embodiments thereof,which description should be considered in conjunction with theaccompanying drawings, which are included for the purpose of exemplaryillustration of various aspects of the present invention to aid indescription, and not for purposes of limiting the invention.

FIG. 1 illustrates exemplary data transmission systems.

FIG. 2 illustrates exemplary data transmission systems.

FIG. 3 illustrates an exemplary plot of Power Normalized Cairns PSWAlphabets.

FIG. 4 illustrates an exemplary plot of Cairns PSW Alphabet with ZeroBoundary Roots.

FIG. 5 illustrates an exemplary plot of Polynomial 1 Roots Before SymbolBoundary Time Edit.

FIG. 6 illustrates an exemplary plot of Polynomial 1 Roots After SymbolBoundary Time Edit.

FIG. 7 illustrates an exemplary plot of a PSW-Gaussian Convolution.

FIG. 8 illustrates an exemplary plot of Fractional Cycle ModulationWaveforms.

FIG. 9 illustrates an exemplary plot of a PSW Alphabet from Offset MCOptimization.

FIG. 10 illustrates an exemplary plot of a PSW Alphabet Convolved withSinc Function Polynomial.

In the drawings and detailed description, the same or similar referencenumbers may identify the same or similar elements. It will beappreciated that the implementations, features, etc., described withrespect to embodiments in specific figures may be implemented withrespect to other embodiments in other figures, unless expressly stated,or otherwise not possible.

DETAILED DESCRIPTION OF THE INVENTION

Aspects of the invention are disclosed in the following description andrelated drawings directed to specific embodiments of the invention.Alternate embodiments may be devised without departing from the spiritor the scope of the invention. Additionally, well-known elements ofexemplary embodiments of the invention will not be described in detailor will be omitted so as not to obscure the relevant details of theinvention. Further, to facilitate an understanding of the descriptiondiscussion of several terms used herein follows.

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration” and not as a limitation. Any embodimentdescribed herein as “exemplary” is not necessarily to be construed aspreferred or advantageous over other embodiments. Likewise, the term“embodiments of the invention” does not require that all embodiments ofthe invention include the discussed feature, advantage or mode ofoperation.

Further, many embodiments are described in terms of sequences of actionsto be performed by, for example, elements of a computing device. It willbe recognized that various actions described herein can be performed byspecific circuits (e.g., application specific integrated circuits(ASICs)), by field programmable gate arrays, by program instructionsbeing executed by one or more processors, or by a combination thereof.Additionally, sequence(s) of actions described herein can be consideredto be embodied entirely within any form of computer readable storagemedium having stored therein a corresponding set of computerinstructions that upon execution would cause an associated processor toperform the functionality described herein. Thus, the various aspects ofthe invention may be embodied in a number of different forms, all ofwhich have been contemplated to be within the scope of the claimedsubject matter. In addition, for each of the embodiments describedherein, the corresponding form of any such embodiments may be describedherein as, for example, “logic configured to” perform the describedaction.

FIG. 1 shows a system 10 including exemplary transmitter 12 and receiver14 pairs that may be used in transmission or communication systems, suchas further shown in FIG. 2 . Bits, usually in the form of data, beinginput in a sequence to the system 10 may be converted to PSWs in aChannel Encoder 16 section of the transmitter 12, as well as have othersignal processing performed to prepare the transmission signal. The PSWsmay then be used to modulate a carrier provided by a carrier source 20using an external modulator 22 as shown in FIG. 1 or to directlymodulate the carrier source 20 to produce the transmission signal.Particularly if implemented in conjunction with ISA transmission, thePSWs may be transmitted using multiple carriers simultaneously. Whilethe Encoder 16 and Decoder 18 are shown as single blocks in FIG. 1 , itwill be appreciated that the Encoder 16 and Decoder 18 may include oneor more stages/components that are used to process the informationpassing through the system, as desired by the skilled artisan.

At the receiver 14, a detector 24 may detect the transmission signal andprovide the transmission signal to signal processors, which may includethe decoder 18 to perform any decoding necessary to output the bits. Thebits output from the system 10 may be in the form of data and clocksignals, or otherwise.

FIG. 2 shows exemplary systems 10 that include a plurality oftransmitters 12 and receivers 14 that may be deployed in varioustransmission and communication systems employing various wired andwireless transmission media 26 and may include PSW technology of thepresent invention. For example, systems 10, such as shown in FIG. 1 andFIG. 2 and other systems, may be deployed in various electrical andoptical wired transmission and communication networks, as well assatellite and terrestrial wireless networks. In various systems, thetransmission signals may be multiplexed in a multiplexer 28 beforetransmission and may require demultiplexing before detection in ademultiplexer 30 after transmission, as is commonly performed in wiredand wireless systems carrying multiple channels.

PSW manipulation technology of the present invention may be implementedin some or all of the transmitters and receivers in a system or networkor only some portion of the transmitters and receivers in the system ornetwork. In this manner, the PSW manipulation technology of the presentinvention may be rolled into new network builds, as well as retrofittedinto existing networks to provide additional capacity in existingnetworks that include transmitters and receivers employing traditionalsignal modulation technology. Transmitters and receivers of the presentinvention may be deployed in existing networks to provide additionalcapacity in networks with unused channels. In addition, one or moretransmitter and receiver pairs of the present invention may be used toreplace traditional transmitters and receivers already deployed inexisting networks to provide additional capacity.

A PSW individually, or a PSW alphabet collectively, can be representedin terms of standard algebraic or Taylor coefficients, as well as interms of its polynomial roots. An infinite number of PSW alphabets arepossible in principle. For illustrative purposes, the first eight Cairnspolynomials (collectively, an 8-symbol PSW alphabet) may be used asdescribed below. See U.S. Pat. No. 10,069,664 for background on theCairns polynomials. The PSWs may be normalized to have the same powerover the symbol time.

FIG. 3 shows an exemplary plot of a normalized Cairns 8-symbol PSWalphabet specified equivalently in terms of (a) standard coefficients,(b) Taylor coefficients and (c) polynomial roots as shown in thefollowing tables (a), (b), and (c).

(a) t{circumflex over ( )}7 t6 t{circumflex over ( )}5 t{circumflex over( )}4 t{circumflex over ( )}3 t{circumflex over ( )}2 t 1 1  2.76737e−06 1.9346e−05  1.1608e−04 5.8039e−04 0.0023 0.0070 0.0139 0.0139 2−2.76737e−06  1.9346e−05 −1.1608e−04 5.8039e−04 −0.0023 0.0070 −0.01390.0139 3 0 −1.8151e−04 0 0.0054 0 −0.0653 0 0.1307 4  −2.8733e−05 00.0012 0 −0.0241 0 0.1448 0 5 0 0 0 −0.0033 0 0 0 0.0794 6 0 0−6.7242e−04 0 0 0 0.0807 0 7 0 −7.0083e−05 0 0 0 0.0252 0 0 8 −9.8749e−06 0 0 0 0.0083 0 0 0

(b) t{circumflex over ( )}7/7! t{circumflex over ( )}6/6! t{circumflexover ( )}5/5! t{circumflex over ( )}4/4! t{circumflex over ( )}3/3!t{circumflex over ( )}2/2! t 1 1 0.0139 0.0139 0.0139 0.0139 0.01390.0139 0.0139 0.0139 2 −0.0139 0.0139 −0.0139 0.0139 −0.0139 0.0139−0.0139 0.0139 3 0 −0.1307 0 0.1307 0 −0.1307 0 0.1307 4 −0.1448 00.1448 0 −0.1448 0 0.1448 0 5 0 0 0 −0.0794 0 0 0 0.0794 6 0 0 −0.0807 00 0 0.0807 0 7 0 0.0505 0 0 0 0.0505 0 0 8 −0.0498 0 0 0 0.0498 0 0 0

(c) Root 1 Root 2 Root 3 Root 4 Root 5 Root 6 Root 7 1 −2.7590 + −2.3799− 1.6290i −2.3799 + −1.1472 − 3.1240i −1.1472 + 1.4066 − 4.2251i1.4066 + 0.0000i 1.6290i 3.1240i 4.2251i 2 2.7590 + 2.3799 − 1.6290i2.3799 + 1.1472 − 3.1240i 1.1472 + −1.4066 − 4.2251i − 1.4066 + 0.0000i1.6290i 3.1240i 4.2251i 3 −1.5699 + 1.5699 + NaN + −3.9281 − 1.2892i−3.9281 + 3.9281− 1.2892i 3.9281 + 0.0000i 0.0000i 0.0000i 1.2892i1.2892i 4 −3.0786 + 0.0000 + 3.0786 + −4.4340 − 1.8438i −4.4340 + 4.4340− 1.8438i 4.4340 + 0.0000i 0.0000i 0.0000i 1.8438i 1.8438i 5 −2.2134 +2.2134 + NaN + NaN + NaN + −0.0000 − 2.2134i 0.0000 + 0.0000i 0.0000i0.0000i 0.0000i 0.0000i 2.2134i 6 −3.3098 + 0.0000 + 3.3098 + NaN +NaN + −0.0000 − 3.3098i 0.0000 + 0.0000i 0.0000i 0.0000i 0.0000i 0.0000i3.3098i 7 −4.3559 + 0.0000 + 0.0000 + 4.3559 + NaN + −0.0000 − 4.3559i−0.0000 + 0.0000i 0.0000i 0.0000i 0.0000i 0.0000i 4.3559i 8 −5.3836 +0.0000 + 0.0000 + 0.0000 + 5.3836 + 0.0000 − 5.3836i 0.0000 + 0.0000i0.0000i 0.0000i 0.0000i 0.0000i 5.3836i

In each of the above tables, the row number corresponds to thepolynomial number in the legend of FIG. 3 . A coefficient representationof a PSW alphabet (table (a): Cairns PSW Alphabet Standard AlgebraicCoefficients or table (b): Cairns PSW Alphabet Taylor Coefficients) maybe necessary for calculating time-sequence amplitude values for symbolwaveforms as part of transmission stream generation. The coefficientrepresentations are also useful for PSW alphabet design.

As further described below, polynomial roots-based PSW alphabetrepresentations (e.g., table (c)) may be also very useful for PSWalphabet design. It may be desirable to be able to switch betweenpolynomial coefficient and polynomial roots representations. Techniquesfor doing so are well-known to the art, and are implemented, forinstance, using the well-known MATLAB® roots and poly functions.

When translating back and forth between PSW alphabet coefficient androots representations, there are two issues to consider, 1) the roots ofa polynomial only specify a particular polynomial to within a constantscaling factor, and 2) different polynomials in the same PSW alphabetmay have different numbers of roots.

With regard to the 1^(st) issue, consider that the roots specify thezero-crossings of a polynomial, but zero crossings are not affected byscale. Consequently, the roots representation of a polynomial does notspecify the scale of the polynomial.

For example, if we know that the roots of a polynomial are 2 and 3, onepossible equation is:

(t−2)(t−3)=0  (3.1)

which can be expanded as

t ²−5t+6=0  (3.2)

Equation 3.1 is not affected by multiplying by a constant k, but thecoefficients are affected by a scaling factor of k. Effectively, theequivalence between polynomial coefficients and polynomial roots is onlyprecise to within a scaling factor. Therefore, where the ability totranslate uniquely back and forth between polynomial coefficients androots representations may be desirable, a “polynomial scaling factor”may be specified and stored for each polynomial in order to create anequivalence between polynomial coefficients and polynomial roots.Generally, this scaling factor may be determined by power normalization.

The second issue to consider is that the polynomials in a PSW alphabetmay not all have the same number of roots, which arises because thenumber of roots is equal to the highest degree of a polynomial's terms.If different polynomials in the PSW alphabet have different highestdegrees, then they will have different numbers of roots.

For instance, in table (b) (Cairns PSW Alphabet Taylor Coefficients),the polynomials corresponding to rows 1, 2, 4 and 8 are seventh degreeand have seven roots; the polynomials corresponding to rows 3 and 7 aresixth degree and have six roots; the polynomial corresponding to row 6is fifth degree and has five roots; and the polynomial corresponding torow 5 is fourth degree and has four roots. Subject to the considerationsin the next paragraph, these observations are readily confirmed byreference to table (c) containing the actual roots.

The possible heterogeneity of the number of roots in a PSW alphabet ismainly a data management issue. In order to store the PSW alphabet rootsin a square array, some entries have to be marked as ‘not roots’. Thismay be done using a special character, such as the MATLAB® ‘Not aNumber’ (NaN), as shown in table (c) for the Cairns PSW Alphabet Roots.Note that the number of ‘NaN’ roots entries that a polynomial has isequal to the difference between the highest degree represented and thedegree of the particular polynomial.

Given the above framework, one or more polynomials in a PSW alphabet maybe modified by changing, or editing, either the polynomial coefficientsor their roots and/or shaping the initial PSW alphabet to produce adesigned PSW alphabet. One application of roots editing is to placepolynomial roots at symbol time boundaries in order to reduceinter-symbol discontinuities. For instance, in FIG. 3 , the Cairns PSWAlphabet Plot, the PSWs have very different amplitudes at both the startand end of the symbol time. In a transmission stream constructed from asequence of these PSWs, this may result in significant inter-symboldiscontinuities and increase the Occupied Bandwidth (OBW) of thetransmitted signal. One of ordinary skill will appreciate that it isgenerally desirable to maximize the bits/bandwidth (i.e., Hertz)transmitted in a transmission system.

An approach to removing zeroth-order Derivative Discontinuities (DDs) isto apply root editing to each polynomial to place a root at the startand end of each symbol time. This forces each PSW to start and end atzero amplitude. While in principle any polynomial roots can be moved tothe symbol boundary times, polynomial modification may be minimized bymoving the roots that were initially closest to the boundary.

The equivalent of FIG. 3 , Cairns PSW Alphabet Plot, with roots editedin this way is shown in FIG. 4 , the Cairns PSW Alphabet with ZeroBoundary Roots.

For illustrative purposes, FIGS. 5 and 6 show the roots of polynomialnumber 1 from the Cairns PSW alphabet table (a) before and after thenearest roots are moved to the symbol time boundaries of −π and π inradians. Note that because the PSWs are real-valued, every root with anon-zero imaginary part must have a matching root which is its complexconjugate (same real value, negative imaginary value). Consequently, ifa complex root is altered, its conjugate root must be altered in thecorresponding way, so that the roots remain conjugates. This may be seenin FIGS. 5 and 6 , where the two conjugate roots on the right both movedto the value of π on the real axis.

In addition to editing polynomial roots, another technique to reduce OBWmay be to convolve PSW coefficients with the polynomial equivalent ofpulse-shaping filters. As an illustrative example, a Gaussianconvolution polynomial may be created with the following MATLAB codefragment, for a specified value of σ and a specified number ofpolynomial terms. The Gaussian mean value may be taken to be zero. Notethat this code coverts the standard Gaussian parameter of x² to anequivalent representation in terms of x in order to facilitateconvolution with PSWs that are expressed in terms of powers of x.

-   -   coeff=1/(2*sigma{circumflex over ( )}2);    -   num_terms_x=num_terms;    -   num_terms_x2=floor(num_terms/2);    -   polyx=zeros(1,num_terms_x);    -   polyx2=zeros(1,num_terms_x2);    -   for idx=0:num_terms_x2−1        -   polyx2(idx+1)=((−1*coeff){circumflex over            ( )}idx)/factorial(idx));        -   polyx(2*idx+1)=polyx2(idx+1);    -   end    -   cony poly=fliplr(polyx)*sqrt(coeff/pi);

FIG. 7 , PSW-Gaussian Convolution, shows the effect of convolving thePSW alphabet shown in FIG. 3 , the Cairns PSW Alphabet Plot, with theGaussian polynomial having σ=1. It can be seen that Gaussian convolutionproduces boundary smoothing, which tends to improve OBW. Convolutiongenerally improves with more terms. In this example, the number of termsin both the PSW alphabet and the Gaussian was taken as 64, and theresult of the convolutions were truncated to 64 terms. The same approachcan be used to apply other filters within the context of PSWs, such asthe raised cosine filter and the root raised cosine filter.

Stochastic Optimization of PSWs

Stochastic optimization methods, which are optimization methods thatgenerate and use random variables, are well-known. However, theirapplication in the telecommunication industry to symbol waveform design,and more particularly to PSW alphabet design, has not previously beendisclosed.

Monte Carlo (MC) optimization is a particular well-known kind ofstochastic optimization; however, MC optimization and other stochasticoptimizations have not previously been applied to PSW alphabet design.Methods for applying MC optimization to PSW alphabet design aredisclosed here for illustrative purposes. However, the application ofother forms of optimization, such as simulated annealing, machinelearning, etc. are also contemplated by this disclosure.

MC optimization may be applied to PSW manipulation to make thepolynomials in a PSW alphabet as distinct from each other as possible.The greater the distinguishability between the PSWs, the easier it willbe for the demodulator to distinguish between them in the presence ofAdditive White Gaussian Noise (AWGN) or other channel impairments, andthus to improve (lower) the Bit Error Rate (BER) performance as afunction of the energy per bit to noise power spectral density ratio(Eb/No).

A useful measure for the distinguishability between two polynomials isthe Root Mean Square (RMS) separation between the values of the twopolynomials at a set of sample points in the symbol time. Otherdistinguishability measures are also possible, but RMS is used here forillustrative purposes.

Thus, an important problem which can be addressed with MC optimizationis to maximize the minimum RMS separation between all pairs ofpolynomials in a PSW alphabet. However, a conflicting constraint is theneed to also control for OBW. Maximizing minimum RMS separation byitself may lead to a PSW alphabet exhibiting high boundary DerivativeDiscontinuities (DDs) or other waveform features that may produce highOBW.

The PSW alphabet MC optimization may be a balance between maximizingpolynomial distinguishability (possibly measured by RMS separation),while minimizing or at least constraining OBW. Exemplary techniques forbalancing these objectives are described for illustrative purposesbelow. However, other similar optimization problems are alsocontemplated by this disclosure, such as the opposite problem ofminimizing PSW alphabet OBW with a PSW alphabet RMS separationconstraint.

The following considerations affect MC optimization implementation ofPSW alphabet optimization:

A number of MC trials, for instance one million, must be specified.

The MC can operate in terms of random variation applied to eitherpolynomial coefficients or polynomial roots.

In each MC trial, the MC optimization can start either from a random PSWalphabet (“fully random” MC), or from a specified input, or initial, PSWalphabet to which the MC optimization generates random offsets. We callthe latter approach “offset MC”.

Effectively, the difference between fully random MC and offset MC isthat in the former random variation is added to zero-valued polynomialcoefficients or roots, and in the latter random variation is added tospecified polynomial coefficients or roots.

Fully random MC may be useful if one has no knowledge of what a good PSWalphabet is likely to be for a particular communication/datatransmission problem. Offset MC may be useful if one has a reasonablecandidate PSW alphabet which one wants to further improve.

On each MC trial, a specified amount of random variation may be added toeach polynomial coefficient or polynomial root. One method of specifyingthe amount of random variation may be in terms of the mean RMS of thePSWs in a PSW alphabet. For instance, if the mean RMS of all polynomialsin a PSW alphabet is R, evaluated at specified sample points across thesymbol time, then the amount of random variation to be added to eachcoefficient or root of each polynomial could be specified (for instance)to be between zero and 0.1*R.

On each MC trial, after adding random variation, the PSW alphabet may bepower normalized. Otherwise, the generated PSW alphabet may generate ahigh minimum RMS separation between any pair of polynomials not becausethe PSW alphabet may be inherently good, but merely because the randomvariation happened to increase its overall power, and hence RMSseparation.

On each MC trial, some constraint check or operation may be necessary.For instance, in order to control for OBW, a constraint may be specifiedthat the symbol time boundary zeroth, first, and/or second DDs betweenany pair of polynomials can be no greater than a particular value. Ifthis constraint is not met, the PSW alphabet generated by the MC trialmay be rejected, regardless of its minimum RMS separation.

As another illustrative example, a constraint may be applied that everypolynomial in the PSW alphabet generated by a MC trial should beconvolved with a pulse-shaping polynomial for OBW control as describedabove. This convolution typically should occur before the minimum RMSseparation calculation. In the case of a constraint such as convolutionwhich may alter power, power normalization typically should occur afterthe constraint is applied.

On each MC trial, a “goodness value” of the generated PSW alphabet mustbe calculated. For illustrative purposes, minimum RMS separation betweenany pair of PSW alphabet polynomials may be considered to be thegoodness value. However, other goodness values are possible andcontemplated by this disclosure.

On each MC trial, if the goodness value exceeds the goodness value ofany previous MC trial, the generated PSW alphabet and its goodness valueare stored as the candidate best solution.

After all MC trials, the best PSW alphabet generated and its goodnessvalue are returned. In the case of offset MC, if the best PSW alphabetgenerated has a lower goodness value than the input PSW alphabet thenthis fact should be noted.

In the case of offset MC, a series of MC trials may be run in which atthe end of each series the best PSW alphabet (if it has a bettergoodness value than the input/initial PSW alphabet) or the input PSWalphabet (otherwise) may be used as the input/initial PSW alphabet forthe next series.

If a series of offset MC optimizations are run, each consisting of anumber of trials, the amount of randomness added in each run may bevaried. In general, initially the amount of randomness should berelatively high, to search a broad PSW alphabet space. As the generatedPSW alphabet goodness values approach a limiting value, the amount ofrandomness should be lowered in order to search more intensely close toa good solution.

If randomization is applied to polynomial roots, care must be taken tomake sure that the polynomial remains real-valued, so that any stream oftime amplitude values produced from the PSW alphabet will bereal-valued. For a polynomial to be real-valued, every complex root musthave a paired complex conjugate. A method for insuring that a polynomialremains real-valued is:

If a root is real, add a real-valued random variation.

If a root is complex, it must have a paired root which is its complexconjugate. Add a complex random offset to the complex root, then adjustthe root's complex conjugate accordingly.

Optionally, some polynomial roots may be specified as ‘fixed’: i.e., theMC optimization is not allowed to alter them. For instances, roots maybe fixed at the symbol time boundaries to ensure that all PSWs generatedwill have zero amplitude at the start and end of each symbol time.

Fractional Cycle Modulation

The technique of constructing a signal from one or more sinusoidalwaves, each of which finishes one or more complete cycles during asymbol time or Transition Time Interval (TTI), is well-known intelecommunications. Notably, Orthogonal Frequency-Division Multiplexing(OFDM) works on this principle.

The present invention includes the use of Fractional Cycle Modulation(FCM), in which sinusoidal symbol waveforms may complete a fraction(which may be less than one) of a complete cycle per symbol time. Anexemplary FCM symbol waveform alphabet is shown in FIG. 8 , FractionalCycle Modulation Waveforms.

FCM has not previously been applied to telecommunications. However, thepresent invention enables FCM to be implemented for data transmission byusing FCM symbol waveforms represented by polynomials and defining PSWalphabets.

In FIG. 8 , Fractional Cycle Modulation Waveforms, two of the symbolwaveforms each complete two cycles over the interval −π to π; two of thesymbol waveforms each complete one cycle over the interval −π to π; andfour of the symbol waveforms each complete one-half cycle over theinterval −π to π.

As with other modulation techniques, such as Phase Shift Keying (PSK) orQuadrature Amplitude Modulation (QAM), FCM communication can be based onassigning a bit string to each symbol waveform. A message may betransmitted by translating bits to symbol waveforms in the encoder orelsewhere in the transmit string, transmitting the symbol waveformsusing the transmitter and by recognizing the sequence of transmittedsymbol waveforms in the receiver and looking up the corresponding bitstrings in the decoder or other processor in the receive string.

In terms of the preceding discussion, a benefit of a FCM PSW alphabetmay be that it has relatively high minimum RMS separation, resulting ingood noise resistance. An accompanying characteristic of the FCM PSWalphabet may be a high OBW, resulting from large discontinuities betweenthe PSWs at the symbol time boundaries. The present invention may makeuse of those FCM characteristics to provide an initial PSW alphabet foroffset MC optimization, as described above. Using an FCM PSW alphabet asthe initial PSW alphabet for offset MC optimization may result in a PSWalphabet similar to the one shown in FIG. 9 , with higher minimum RMSseparation.

The PSW alphabet depicted in FIG. 9 , PSW Alphabet from MC Optimization,can be shaped for OBW reduction or for other reasons by polynomialconvolution, as disclosed above. For instance, the result of convolvingthe PSW alphabet shown in FIG. 9 , PSW Alphabet from MC Optimization,with a polynomial version of the sinc function is shown in FIG. 10 , PSWAlphabet Convolved with Sinc Function Polynomial. Notice that zerothorder DDs have been removed by the convolution and the PSW alphabet hasbeen generally smoothed, which improves OBW performance.

In operation, a user of the present invention may perform PolynomialSymbol Waveform (PSW) design via or based on one of the methodsdescribed above to create a manipulated PSW alphabet with a power-of-twonumber of polynomials. As with traditional modulation, each polynomialin the alphabet will correspond to a bit sequence whose length is thelog-base-two of the size of the alphabet.

The user will load the manipulated PSW alphabet into a channelencoder/modulator or other processor that will be used to convert aninput bit sequence into an equivalent sequence of PSWs and into adecoder/demodulator that will convert the PSW sequence back into the bitsequence. The polynomials may be sampled to produce a time amplitudesequence (i.e., stream) for transmission.

In various embodiments, the PSW alphabet may be loaded into thetransmitters and receivers as software in a software defined radio(“SDR”) configuration. However, it will be appreciated that the PSWalphabet may be implemented via hardware and/or software that employFPGAs, ASICs, CPUs, etc.

During the PSW alphabet design phase, the user may simulate transmissionof the stream with or without channel impairments, such AWGN, to performOccupied Bandwidth (OBW) measurements as a function of variousparameters, such as Bit Error Rate (BER) and Energy per bit to Noisepower (Eb/No).

The stream is transmitted and then received by the receiver. Thereceived stream is then decoded/demodulated using the PSW alphabetloaded into the receiver to reproduce the bit sequence provided to thetransmitter.

The foregoing description and accompanying drawings illustrate theprinciples, preferred embodiments and modes of operation of theinvention. However, the invention should not be construed as beinglimited to the particular embodiments discussed above. Additionalvariations of the embodiments discussed above will be appreciated bythose skilled in the art.

Therefore, the above-described embodiments should be regarded asillustrative rather than restrictive. Accordingly, it should beappreciated that variations to those embodiments can be made by thoseskilled in the art without departing from the scope of the invention asdefined by the following claims.

What is claimed is:
 1. A system comprising: at least one transmitter toreceive an input bit sequence, convert the input bit sequence into asequence of polynomial symbol waveforms (PSWs) selected from a designedPSW alphabet, the designed PSW alphabet being formed by providing aninitial PSW alphabet having polynomial coefficients and polynomialroots, modifying at least one of the polynomial coefficients orpolynomial roots of the initial PSW alphabet produce to the designed PSWalphabet, and transmit the sequence of polynomial symbol waveforms; andat least one receiver to receive the sequence of polynomial symbolwaveforms, convert the received sequence of polynomial symbol waveformsinto an output bit sequence based on the designed PSW alphabet, andoutput the output bit sequence.
 2. The system of claim 1, wheremodifying the PSW alphabet includes: shaping one of the edited PSWalphabet and initial PSW alphabet to produce the designed PSW alphabet.3. The system of claim 1, where modifying the PSW alphabet includes atleast one of: translating at least one polynomial root from one or morePSWs of the PSW alphabet from a starting location to an end location inthe complex plane; adjusting complex conjugates of translated roots thatare complex to keep the polynomial real-valued; shaping at least one ofthe translated and adjusted PSWs using polynomial convolution to providethe designed PSW alphabet.
 4. The system of claim 3, where the endlocation for the translated polynomial roots are at the symbol timeboundaries with amplitude zero.
 5. The system of claim 3, where thepolynomial convolution is performed using one of raised cosine, rootraised cosine, Gaussian, or other band-limiting or pulse-shapingfilters.
 6. The system of claim 1, where modifying includes applyingrandom variation to at least one of the polynomial coefficients andpolynomial roots of an initial PSW alphabet to produce an edited PSWalphabet; calculating a goodness measure for the initial and edited PSWalphabets; comparing the goodness measure of the initial PSW alphabet tothe goodness measure of the edited PSW alphabet; and setting the initialPSW alphabet equal to the edited PSW alphabet when the edited PSWalphabet has a higher goodness measure.
 7. The system of claim 6, wherethe random variation is determined by at least one of Monte Carlooptimization and machine learning.
 8. The system of claim 6, furthercomprising performing a power normalization of the edited PSW alphabetafter applying the random variation.
 9. The system of claim 6, furthercomprising setting the initial PSW alphabet to the edited PSW alphabetwhen the goodness measure of the edited PSW alphabet is better than thegoodness measure of the initial PSW alphabet.
 10. The system of claim 9,where the initial PSW alphabet is modified for at least one of a numberof repetitions, until a goodness measure is achieved, and until agoodness measure is maximized.
 11. The system of claim 6, where thegoodness measure for a PSW alphabet is based on calculating the minimumRoot Mean Square (RMS) separation between all pairs of PSWs with ahigher minimum RMS separation being interpreted as a higher goodnessmeasure.
 12. The system of claim 1, where the initial PSW alphabet isprovided using fractional cycle modulation to generate the PSWs.
 13. Anon-transitory computer readable medium storing instructions, theinstructions comprising: one or more instructions which, when executedby one or more processors, cause the one or more processors to modifythe initial PSW alphabet as provided in claim
 1. 14. A non-transitorycomputer readable medium storing instructions, the instructionscomprising: one or more instructions which, when executed by one or moreprocessors, cause the one or more processors to: convert, by atransmitter, an input bit sequence into a sequence of polynomial symbolwaveforms (PSWs) selected from a designed PSW alphabet, the designed PSWalphabet being formed by providing an initial PSW alphabet havingpolynomial coefficients and polynomial roots, modifying at least one ofthe polynomial coefficients or polynomial roots of the initial PSWalphabet produce to the designed PSW alphabet; and transmit, by thetransmitter, the sequence of polynomial symbol waveforms.
 15. A systemcomprising: a transmitter to receive an input bit sequence, convert theinput bit sequence into a sequence of polynomial symbol waveforms (PSWs)selected from a designed PSW alphabet, the designed PSW alphabet beingformed by providing an initial PSW alphabet having polynomialcoefficients and polynomial roots, modifying at least one of thepolynomial coefficients or polynomial roots of the initial PSW alphabetproduce to the designed PSW alphabet, and transmit the sequence ofpolynomial symbol waveforms.
 16. The system of claim 15, furthercomprising: a receiver to receive the sequence of polynomial symbolwaveforms, convert the received sequence of polynomial symbol waveformsinto an output bit sequence based on the designed PSW alphabet, andoutput the output bit sequence.
 17. The system of claim 15, where theinitial polynomial symbol waveform alphabet is a random PSW alphabet.18. The system of claim 15, where the initial polynomial symbol waveformalphabet is a SPDM PSW alphabet.
 19. The system of claim 15, where theinitial polynomial symbol waveform alphabet is modified to maximize RMSseparation between the polynomial symbol waveforms.
 20. The system ofclaim 15, where the initial polynomial symbol waveform alphabet ismodified using instantaneous spectral analysis.